#395452

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62853	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,17,19,12)(8,21,22,13)(9,24)(10,15)(11,27)(14,25)(16,26)(18,33)(20,32,28,36)(23,34,29,31)(30,35);(1,2)(3,4)(5,7)(6,12)(8,9)(10,19)(11,28)(13,33)(14,31)(15,17)(16,36)(18,22)(20,26)(21,24)(23,30)(25,29)(27,32)(34,35);(2,4)(5,8)(6,13)(7,18)(9,19)(10,22)(11,29)(12,24)(14,32)(15,21)(16,31)(17,33)(20,25)(23,26)(27,34)(28,30)(35,36);(5,9,10,18)(6,14,15,35)(7,8,19,22)(11,30,26,25)(12,31,17,34)(13,32,21,36)(16,33,27,24)(20,29,28,23)

Polycyclic presentation:

2^9|a^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302648);